Neumann–Neumann methods

In mathematics, Neumann–Neumann methods are domain decomposition preconditioners named so because they solve a Neumann problem on each subdomain on both sides of the interface between the subdomains.[1] Just like all domain decomposition methods, so that the number of iterations does not grow with the number of subdomains, Neumann–Neumann methods require the solution of a coarse problem to provide global communication. The balancing domain decomposition is a Neumann–Neumann method with a special kind of coarse problem.

More specifically, consider a domain Ω, on which we wish to solve the Poisson equation

− Δ u = f , u | ∂ Ω = 0 {\displaystyle -\Delta u=f,\qquad u|_{\partial \Omega }=0}

for some function f. Split the domain into two non-overlapping subdomains Ω1 and Ω2 with common boundary Γ and let u1 and u2 be the values of u in each subdomain. At the interface between the two subdomains, the two solutions must satisfy the matching conditions

u 1 = u 2 , ∂ n 1 u 1 = ∂ n 2 u 2 {\displaystyle u_{1}=u_{2},\qquad \partial _{n_{1}}u_{1}=\partial _{n_{2}}u_{2}}

where  n i {\textstyle n_{i}} is the unit normal vector to Γ in each subdomain.

An iterative method with iterations k = 0, 1, ... for the approximation of each ui (i = 1, 2) that satisfies the matching conditions is to first solve the Dirichlet problems

− Δ u i ( k ) = f i in Ω i , u i ( k ) | ∂ Ω = 0 , u i ( k ) | Γ = λ ( k ) {\displaystyle {\begin{aligned}-&\Delta u_{i}^{(k)}=f_{i}~~{\text{in}}~~\Omega _{i},\\\\[1.3ex]&\left.u_{i}^{(k)}\right|_{\partial \Omega }=0,\quad \left.u_{i}^{(k)}\right|_{\Gamma }=\lambda ^{(k)}\end{aligned}}}

for some function λ(k) on Γ, where λ(0) is any inexpensive initial guess. We then solve the two Neumann problems

− Δ ψ i ( k ) = 0 in Ω i , ψ i ( k ) | ∂ Ω = 0 , ∂ n i ψ i ( k ) | Γ = ω ( ∂ n 1 u 1 ( k ) + ∂ n 2 u 2 ( k ) ) . {\displaystyle {\begin{aligned}-&\Delta \psi _{i}^{(k)}=0~~{\text{in}}~\Omega _{i},\\\\[1.3ex]&\left.\psi _{i}^{(k)}\right|_{\partial \Omega }=0,\quad \left.\partial _{n_{i}}\psi _{i}^{(k)}\right|_{\Gamma }=\omega \left(\partial _{n_{1}}u_{1}^{(k)}+\partial _{n_{2}}u_{2}^{(k)}\right).\end{aligned}}}

We then obtain the next iterate by setting

λ ( k + 1 ) = λ ( k ) − ω ( θ 1 ψ 1 ( k ) + θ 2 ψ 2 ( k ) ) on Γ {\displaystyle \lambda ^{(k+1)}=\lambda ^{(k)}-\omega \left(\theta _{1}\psi _{1}^{(k)}+\theta _{2}\psi _{2}^{(k)}\right)~~{\text{on}}~\Gamma }

for some parameters ω, θ1 and θ2.

This procedure can be viewed as a Richardson iteration for the iterative solution of the equations arising from the Schur complement method.[2]

This continuous iteration can be discretized by the finite element method and then solved—in parallel—on a computer. The extension to more subdomains is straightforward, but using this method as stated as a preconditioner for the Schur complement system is not scalable with the number of subdomains; hence the need for a global coarse solve.

## See also

[edit]

* Neumann–Dirichlet method

## References

[edit]

1. ^ A. Klawonn and O. B. Widlund, FETI and Neumann–Neumann iterative substructuring methods: connections and new results, Comm. Pure Appl. Math., 54 (2001), pp. 57–90.
2. ^ A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications 1999.

* v
* t
* e

Numerical methods for partial differential equations

Finite difference|

| Parabolic|

* Forward-time central-space (FTCS)
* Crank–Nicolson

|

Hyperbolic|

* Lax–Friedrichs
* Lax–Wendroff
* MacCormack
* Upwind
* Method of characteristics

Others|

* Alternating direction-implicit (ADI)
* Finite-difference frequency-domain (FDFD)
* Finite-difference time-domain (FDTD)

Finite volume|

* Godunov
* High-resolution
* Monotonic upstream-centered (MUSCL)
* Advection upstream-splitting (AUSM)
* Riemann solver
* Essentially non-oscillatory (ENO)
* Weighted essentially non-oscillatory (WENO)

Finite element|

* hp-FEM
* Extended (XFEM)
* Discontinuous Galerkin (DG)
* Spectral element (SEM)
* Mortar
* Gradient discretisation (GDM)
* Loubignac iteration
* Smoothed (S-FEM)

Meshless/Meshfree|

* Smoothed-particle hydrodynamics (SPH)
* Peridynamics (PD)
* Moving particle semi-implicit method (MPS)
* Material point method (MPM)
* Particle-in-cell (PIC)

Domain decomposition|

* Schur complement
* Fictitious domain
* Schwarz alternating
* additive
* abstract additive
* Neumann–Dirichlet
* Neumann–Neumann
* Poincaré–Steklov operator
* Balancing (BDD)
* Balancing by constraints (BDDC)
* Tearing and interconnect (FETI)
* FETI-DP

Others|

* Spectral
* Pseudospectral (DVR)
* Method of lines
* Multigrid
* Collocation
* Level-set
* Boundary element
* Method of moments
* Immersed boundary
* Analytic element
* Isogeometric analysis
* Infinite difference method
* Infinite element method
* Galerkin method
* Petrov–Galerkin method
* Validated numerics
* Computer-assisted proof
* Integrable algorithm
* Method of fundamental solutions

Related|

* Numerical methods for ordinary differential equations
* Numerical integration

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template